In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible...

In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a counital coassociative coalgebra...

counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups. Many quantum groups...

Lie algebra Lie-* algebra Lie algebra bundle Lie algebra cohomology Lie algebra representation Lie algebroid Lie bialgebra Lie coalgebra Lie conformal algebra...

reductive Lie algebra. There is an equivalence of categories between finite-dimensional Manin triples and finite-dimensional Lie bialgebras. More precisely...

a Lie algebra Leibniz algebra Lie algebra cohomology Lie algebra extension Lie algebra representation Lie bialgebra Lie coalgebra Lie n-algebra Lie operad...

Poisson-Lie groups and finite-dimensional Lie bialgebras, extending the classical equivalence between simply connected Lie groups and finite-dimensional Lie algebras...

× V → F. Bialgebra: an associative algebra with a compatible coalgebra structure. Lie bialgebra: a Lie algebra with a compatible bialgebra structure...

vector bundles. Lie bialgebroids are the vector bundle version of Lie bialgebras. A Lie algebroid consists of a bilinear skew-symmetric operation [ ⋅ , ⋅...

Euclidean group E(3) of motions in 3 dimensions. Hopf algebra Lie bialgebra Poisson–Lie group Quantum affine algebra Schwiebert, Christian (1994), Generalized...

)} is a pre-Lie algebra. Lie coalgebra Lie bialgebra Lie algebra cohomology Frobenius algebra Quasi-Frobenius ring Jacobson, Nathan, Lie algebras, Republication...

concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure...

statistical mechanics and topology. Yang–Baxter equation Hopf algebra Lie bialgebra Yangian Braid theory Quantum groups Baxter, R. (1982). "Exactly solved...

trivially holds at orders ℏ 0 , ℏ {\displaystyle \hbar ^{0},\hbar } ). Lie bialgebra Yangian Reidemeister move Quasitriangular Hopf algebra Yang–Baxter operator...

coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain...

include the tensor algebra, the exterior algebra, Hopf algebras and Lie bialgebras. Unlike the polynomial case above, none of these are commutative. Therefore...

application to the quantum Yang–Baxter equation, the quantisation of Lie bialgebras and quantum Lévy area. Hudson, R. L.; Ion, P. D. F.; K. R. Parthasarathy...

⁠. The exterior algebra (as well as the symmetric algebra) inherits a bialgebra structure, and, indeed, a Hopf algebra structure, from the tensor algebra...

into a sum of solitons and a decaying dispersive part. Lax pair Lie bialgebra Poisson–Lie group Krüger, Helge; Teschl, Gerald (2009), "Long-time asymptotics...

comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the...

satisfying these conditions. This is the motivation for the definition of a bialgebra, where Δ is called the comultiplication and ε is called the counit. In...

Its formal group ring (also called its hyperalgebra or its covariant bialgebra) is a cocommutative Hopf algebra H constructed as follows. As an R-module...

N=4 Supersymmetric Yang–Mills". YouTube. 30 November 2021. "Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction by Niklas Beisert"...

notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative...

applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. Generally, the index set of a graded ring is assumed to be the set...

right module over R can be considered a left module over Rop. Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra...

(mathematics) Category theory Monoidal category Groupoid Group object Coalgebra Bialgebra Hopf algebra Magma object Torsion (algebra) Symbolic mathematics Finite...

lifts, given the prescription above. See, however, the discussion of the bialgebra structure in the article on tensor algebras for a review of some of the...

operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra Hopf algebra v t e...

functor in monoidal categories. Commutator Non-associative algebra Quasi-bialgebra – discusses the Drinfeld associator Bremner, M.; Hentzel, I. (March 2002)...